Exercise 3, § 14

a) Let K be a commutative field, A a K-algebra, and B a central simple subalgebra of A of finite rank. Let $B'$ be the commutant of B in A. Prove that the canonical homomorphism from $B\otimes_KB'$ to A is bijective. (Note that B and $B'$ are linearly disjoint over K (VIII, p. 225, Exercise 5, d)). Prove, furthermore, that A is the direct sum of a family $(M_{\iota})_{\iota\in I}$ of $(B\otimes_KB^o$)-modules isomorphic to B, and observe that for every $\iota \in I$, the element of $M_{\iota}$ corresponding to $1\in B$ by such an isomorphism belongs to $B'.)$

b) Conversely, let B be a K-algebra such that for every algebra A containing B and having the same unit element as B, the canonical homomorphism from $B\otimes_KB'$ to A is bijective. Prove that the algebra B is central simple and of finite rank (take A = End$_K(B)$; deduce from the assumption that B is central and quasi-simple, and use Exercise 5, e) of VIII, p. 225).